Resource scheduling is a special class of optimization problem, in which a mix of available resources are utilized to satisfy demand side requirements over a given time horizon at prescribed temporal or spatial resolutions. The decisions to be made involve the determination of optimal operation schedules for all resources involved. The operation schedule for a resource can be described by the startup and shutdown, and utilization at each time step over the scheduling horizon. The operation schedules for all the resources are determined such that various local and global constraints are satisfied and some generalized cost function is minimized. A well-known classic example of resource scheduling problem is the so called unit commitment problem in the electric power industry, where the resources are the thermal, hydro power plants in the system; the demand side requirements are the total customer hourly load over 24 or 168 hours. The startup, shutdown, and utilization of each plant is referred to as commitment, de-commitment. and dispatch. The operation schedule for each plant must satisfy the local constraints such as minimum up time, minimum down time, ramping constraint, and available capacity limitations. Collectively, all the plants committed must also satisfy global constraints such as various reserve requirements and energy balance constraints.
The resource scheduling problem, by its combinatorial nature, is very hard. The computational burden increases exponentially with the number of resources and the time steps in the scheduling horizon. To overcome the “curse of dimensionality”, decomposition techniques based on Lagrangian relaxation theory are generally used. In these decomposition methods, the original optimization problem is relaxed by removing the so called “complicating constraints”, also known as “coupling constraints”, to obtain a separable optimization problem, which can be divided into many smaller independent optimization problems, usually referred to as sub-problems. All the sub-problems are solved, e.g., one sub-problem for each unit in the unit commitment problem, and the Lagrangian multipliers are updated at a high level. The solutions of the sub-problems are coordinated by a set of price signals for the complicating constraints.
The performance of a resource scheduling method is measured by the following criteria: (1) optimality, (2) feasibility and (3) speed. Optimality measures how close the solution is to the theoretical best. A feasibility check ensures that all constraints have been satisfied. Speed is how fast the method is at finding the solution. The optimality of the solution is usually measured by the solution gap. The solution gap is a conservative estimate of the closeness of a solution to a theoretical optimum solution in terms of an objective value. Solution gap is defined by (OBJ−LB)/OBJ expressed as a percent, where OBJ is the best integer solution found and LB is the tightest lower bound known for the problem.
An optimization method based on Lagrangian relaxation typically includes two main steps: (1) Lagrangian dual optimization to get an optimal dual solution; and (2) construction of a primal feasible solution. The problem encountered in the first step is an optimization of the non-differentiable dual function, which is commonly solved by variations of subgradient (SG) and cutting plane (CP) methods. This is an iterative process plagued by slow convergence, requiring hundreds, sometimes thousands of iterations. Quickly getting to the optimal dual solution is an attractive and challenging objective in the design of Lagrangian relaxation-based algorithms.
The primal solution corresponding to the optimal dual solution in general is not ensured to be feasible for all the coupling constraints. The second step attempts to make the primal solution feasible by applying various problem dependent heuristics to adjust the primal solution until it becomes feasible. The heuristics rules are generally constructed from experience and insight into the specific problems at hand and lack generality. The method disclosed herein offers a huge performance advantage over the traditional Lagrangian relaxation-based methods by providing fast dual optimization and a problem independent process for feasibility enforcement.